Watkins Willis
06/14/2024 · Junior High School
Find the derivative of the function using the definition of derivativ \[ G(t)=\frac{1-2 t}{4+t} \]
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Step-by-step Solution
Find the first order derivative with respect to \( t \) for \( \frac{1-2t}{4+t} \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dt}\left(\frac{1-2t}{4+t}\right)\)
- step1: Use differentiation rules:
\(\frac{\frac{d}{dt}\left(1-2t\right)\times \left(4+t\right)-\left(1-2t\right)\times \frac{d}{dt}\left(4+t\right)}{\left(4+t\right)^{2}}\)
- step2: Calculate:
\(\frac{-2\left(4+t\right)-\left(1-2t\right)\times 1}{\left(4+t\right)^{2}}\)
- step3: Calculate:
\(\frac{-2\left(4+t\right)-\left(1-2t\right)}{\left(4+t\right)^{2}}\)
- step4: Calculate:
\(\frac{-9}{\left(4+t\right)^{2}}\)
- step5: Rewrite the expression:
\(-\frac{9}{\left(4+t\right)^{2}}\)
- step6: Calculate:
\(-\frac{9}{16+8t+t^{2}}\)
The derivative of the function \( G(t)=\frac{1-2t}{4+t} \) using the definition of the derivative is \(-\frac{9}{16+8t+t^{2}}\).
Quick Answer
The derivative of \( G(t)=\frac{1-2t}{4+t} \) is \(-\frac{9}{16+8t+t^{2}}\).
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